A vibration damper of this type is disclosed in German Offenlegungsschrift DE-A1 3 146 679. This publication describes an elastic engine support consisting of a housing formed by a shell part and a base part, the base part likewise being shell-shaped. A membrane is stretched in the housing by clamping said membrane between the shell part and the base part. This creates a pressure cavity between the membrane and the base part, by means of which pressure cavity air damping is created for vibrations transmitted to the membrane by means of the fixing pin. The membrane itself is made of rubber and likewise contributes to the vibration damping. The fixing pin is fixed to the membrane by means of a plate which is partially enclosed within the membrane and the function of which is exclusively to transmit forces exerted on the connecting pin to the membrane.